(*
  DaoMath Quantum Mechanics
  量子力学的形式化
*)

Require Import Reals.
Require Import Complex.
Require Import Psatz.
Open Scope R_scope.

(** * Hilbert 空间（简化） *)

(** 量子态作为复数向量 *)
Parameter QuantumState : Type.

(** 内积 *)
Parameter inner_product : QuantumState -> QuantumState -> Complex.

(** 内积公理 *)
Axiom inner_positive : forall psi : QuantumState,
  real_part (inner_product psi psi) >= 0.

Axiom inner_conjugate_symmetric : forall psi phi : QuantumState,
  inner_product psi phi = C_conjugate (inner_product phi psi).

Axiom inner_linear_left : forall psi phi chi : QuantumState,
  forall alpha : Complex,
  inner_product (state_plus (state_scale alpha psi) phi) chi =
  C_plus (C_mult alpha (inner_product psi chi))
         (inner_product phi chi)
where "state_plus" := (fun psi phi => psi) (* 占位符 *)
and "state_scale" := (fun alpha psi => psi). (* 占位符 *)

(** * 归一化 *)

Definition is_normalized (psi : QuantumState) : Prop :=
  inner_product psi psi = mkComplex 1 0.

(** * 量子算符 *)

Parameter Operator : Type.
Parameter apply_operator : Operator -> QuantumState -> QuantumState.

(** Hermitian 算符 *)
Definition is_hermitian (A : Operator) : Prop :=
  forall psi phi : QuantumState,
  inner_product (apply_operator A psi) phi =
  C_conjugate (inner_product psi (apply_operator A phi)).

(** Unitary 算符 *)
Definition is_unitary (U : Operator) : Prop :=
  forall psi : QuantumState,
  is_normalized psi ->
  is_normalized (apply_operator U psi).

(** * 测量公理 *)

(** Born 规则 *)
Parameter measurement_probability : 
  QuantumState -> Operator -> R.

Axiom born_rule : forall (psi : QuantumState) (A : Operator),
  is_normalized psi ->
  is_hermitian A ->
  measurement_probability psi A >= 0 /\
  measurement_probability psi A <= 1.

(** 测量后坍缩 *)
Parameter collapse : QuantumState -> Operator -> R -> QuantumState.

Axiom collapse_normalized : 
  forall (psi : QuantumState) (A : Operator) (eigenvalue : R),
  is_normalized psi ->
  is_normalized (collapse psi A eigenvalue).

(** * 不确定性原理 *)

(** 方差 *)
Parameter variance : QuantumState -> Operator -> R.

(** Heisenberg 不确定性原理 *)
Theorem heisenberg_uncertainty :
  forall (psi : QuantumState) (A B : Operator),
  is_normalized psi ->
  is_hermitian A ->
  is_hermitian B ->
  exists delta_A delta_B : R,
  delta_A = sqrt (variance psi A) /\
  delta_B = sqrt (variance psi B) /\
  delta_A * delta_B >= 0. (* 简化版本 *)
Proof.
  intros psi A B H_norm H_A H_B.
  exists (sqrt (variance psi A)).
  exists (sqrt (variance psi B)).
  repeat split.
  - apply sqrt_positivity.
    admit. (* variance is non-negative *)
  - apply sqrt_positivity.
    admit.
  - apply Rmult_le_pos; apply sqrt_pos.
Admitted.

(** * 量子纠缠 *)

(** 张量积（抽象） *)
Parameter tensor_product : QuantumState -> QuantumState -> QuantumState.

Notation "psi ⊗ phi" := (tensor_product psi phi) (at level 40).

(** 可分离性 *)
Definition is_separable (psi : QuantumState) : Prop :=
  exists phi chi : QuantumState,
  psi = phi ⊗ chi.

(** Bell 态（不可分离） *)
Parameter bell_state : QuantumState.

Axiom bell_state_entangled : ~ is_separable bell_state.

(** 定理: Bell 态是归一化的 *)
Axiom bell_state_normalized : is_normalized bell_state.

(** * EPR 佯谬与非局域性 *)

(** 测量结果的关联 *)
Parameter correlation : 
  QuantumState -> Operator -> Operator -> R.

(** Bell 不等式违背 *)
Theorem bell_inequality_violation :
  exists (psi : QuantumState) (A B : Operator),
  is_normalized psi ->
  is_hermitian A ->
  is_hermitian B ->
  Rabs (correlation psi A B) > 1/sqrt(2).
Proof.
  (* 需要具体的 Bell 态和算符 *)
Admitted.

